Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

摘要

We consider the nonlinear Sturm–Liouville problem 1 $$ -u''(t) + f(u(t)) = lambda u(t),quad u(t) > 0, quad t in I := (0, 1),quad u(0) = u(1) = 0, $$ where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R + × L 2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction u α with ‖ u α ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ? f(u)/u p satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ～ α p?1 h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by lim u → ∞ uh′(u).